Question

A debt of $9000 is to be amortized with 8 equal semiannual

payments. If the interest rate is 13%, compounded semiannually,
what is the size of each payment? (Round your answer to the nearest
cent.)

Answer 1

Each semiannual payment is approximately $1,838.53.

Step-by-step explanation:

To calculate the semiannual payment for the amortization of the debt, we can use the formula for the present value of an annuity:

`\[ PVA = P * \left(1 - (1)/((1 + r)^(nt))\right) / r \]`

Where:

- \( P \) is the periodic payment,

- \( r \) is the interest rate per period,

- \( n \) is the total number of periods (in this case, semiannual payments),

- \( t \) is the total number of years.

In this scenario:

-
`\( PVA \)`is the present value of the debt, which is $9,000,

-
`\( r \)`is the interest rate per period (13% compounded semiannually, so \
`( r = 0.065 \)),`

-
`\( n \`) is the total number of semiannual periods (8 payments in total, so
`\( n = 8 \)),`

-
`\( t \)` is the total number of years (1 year for 8 semiannual periods, so
`\( t = 1 \)).`

Substitute these values into the formula and solve for \( P \) to find the semiannual payment.

`\[ 9000 = P * \left(1 - (1)/((1 + 0.065)^(8))\right) / 0.065 \]`

`\[ P \approx (9000)/(\left(1 - (1)/((1 + 0.065)^(8))\right) / 0.065) \]`

`\[ P \approx 1838.53 \]`

Therefore, each semiannual payment is approximately $1,838.53.